Expert: Paul Klarreich - 8/21/2011

Question
QUESTION: Hello! I was wondering if you could clarify whether or not my answers are correct. I don't want to turn them in, and then wind up having to do them all over.

Problem 1:
Write the statement in symbolic form and construct a truth table:
It is false that Wanda Garner is the president or that Judy Ackerman is the treasurer. (I attached the photo of the table I made. Sorry if it looks kind of messy.)

Problem 2:
Use De Morgan’s laws to determine whether the two statements are equivalent:
∼p ∨ ∼q, ∼(p ∧ q)

My answer: ∼p ∨ ∼q = ∼(p ∧ q)
(Not P) or (Not Q) = Not (P and Q)
They are equivalent

Now, on Problem 3, I need a little assistance:
Determine the truth value of the statement when p is T, q is F, and r is F: (p ∨ q) ↔ (∼r ∧ p)

Could you help me, please? Thanks!

ANSWER: Questioner:Miss Susan Wilson
Country:Texas, United States
Category:Advanced Math
Private:No
Question: Hello! I was wondering if you could clarify whether or not my answers are
correct. I don't want to turn them in, and then wind up having to do them all
over.

Problem 1:
Write the statement in symbolic form and construct a truth table:
It is false that Wanda Garner is the president or that Judy Ackerman is the
treasurer. (I attached the photo of the table I made. Sorry if it looks kind of
messy.)

>>>>>>>>> The phrasing is a little vague, but I would interpret it to be:

- (p + q)  

Sorry, I can't make those fancy symbols.  I use:

-p     for  NOT
p + q  for  OR, and
pq     for  AND, and
p -> q for  IMPLIES.

And this sentence has a different TT:

p  q  p+q  -(p+q)
------------------
T  T   T      F
T  F   T      F
F  T   T      F
F  F   F      T


Problem 2:
Use De Morgan’s laws to determine whether the two statements are equivalent:
∼p ∨ ∼q, ∼(p ∧ q)

My answer: ∼p ∨ ∼q = ∼(p ∧ q)
(Not P) or (Not Q) = Not (P and Q)
They are equivalent.

>>>>>>>>>>>>>  yes, that is so.  However, I think you might be expected to work out the TT's for these to actually prove it.  The question said 'determine', not 'guess', so your teacher might not give full credit for this answer. (I wouldn't)

Now, on Problem 3, I need a little assistance:
Determine the truth value of the statement when p is T, q is F, and r is F:

(p ∨ q) ↔ (∼r ∧ p)

(T ∨ F) ↔ (∼F ∧ T)

(T) ↔ (T ∧ T)

(T) ↔ (T)

T


---------- FOLLOW-UP ----------

QUESTION: Thank you so much, Paul, because I am not good with this stuff. Would you mind taking a look at 2 more problems. One I have figured out (I think), and the other, I do not.

Problem 1:
Construct a truth table for (p Λ ~q) ↔ q

My answer:
p       ~q        (p^~q)        q     (p^~q)<->q
T        T          T          F          F
T        F          F          T          F
F        T          F          F          T
F        F          F          T          F

Problem 2:
Construct a truth table for ~q Λ p (Could you help me a little with this)

Answer
Questioner:Miss Susan Wilson
Country:Texas, United States
Category:Advanced Math
Private:No
Subject:Symbolic logic

QUESTION: Thank you so much, Paul, because I am not good with this stuff. Would you mind taking a look at 2 more problems. One I have figured out (I think), and the other, I do not.

Problem 1:
Construct a truth table for (p Λ ~q) ↔ q

My answer:
p       ~q        (p^~q)        q     (p^~q)<->q
T        T          T          F          F
T        F          F          T          F
F        T          F          F          T
F        F          F          T          F

>>>>>>>>>>>>>>>>>  You should ALWAYS start your TT with  p, q  as the first two columns.  Then I suggest (as I did with my own classes) that you number the columns to keep things straight:
1.   2.     3.  ~ 1    4. 1 Λ 3      5. 4 = 2
p  |  q  |  ~q       |   p Λ ~q   |  (p Λ ~q) ↔ q
---+-----+-----------+------------+---------------+
T     T
T     F
F     T
F     F


Problem 2:
Construct a truth table for ~q Λ p (Could you help me a little with this)

>>>>>>>>>>> This should be easy:

p  |  q  |   -q   |  -q and p
---+-----+--------+------------+
T     T
T     F
F     T
F     F

You can fill in the rest.